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5) (a + b - 10)(a - b + 4)
7) x = ± 2
5) Solution :
=> a² - 6a - 40 + 14b - b²
=> a² - 6a + 9 - 49 + 14b - b²
=> (a² - 6a + 9) - (b² - 14b + 49)
=> (a² - 2×a×3 + 3²) - (b² - 2×b×7 + 7²)
=> (a - 3)² - (b - 7)²
=> [ (a - 3) + (b - 7) ] × [ (a - 3) - (b - 7) ]
=> [ a - 3 + b - 7 ] × [ a - 3 - b + 7 ]
=> (a + b - 10)(a - b + 4)
7) Solution :
We have ;
3^x + 1/3^x = 9 1⁄9
Let 3^x = y , then the given equation will be reduced to ; y + 1/y = 9 1⁄9
=> y + 1/y = 9 1⁄9
=> (y² + 1)/y = 82/9
=> 9(y² + 1) = 82y
=> 9y² + 9 = 82y
=> 9y² - 82y + 9 = 0
=> 9y² - 81y - y + 9 = 0
=> 9y(y - 9) - (y - 9) = 0
=> (y - 9)(9y - 1) = 0
=> Either y - 9 = 0 or 9y - 1 = 0
Case1 : y - 9 = 0
=> y - 9 = 0
=> y = 9
=> 3^x = 9
=> 3^x = 3^2
=> x = 2
Case2 : 9y - 1 = 0
=> 9y - 1 = 0
=> 9y = 1
=> y = 1/9
=> 3^x = 3^(-2)
=> x = -2
Hence , x = ± 2 .
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